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A certain kernel (sometimes called the Pick kernel) associated to Schurfunctions on the disk is always positive semi-definite. A generalization ofthis fact is well-known for Schur functions on the polydisk. In this article,we show that the Pick kernel on the polydisk has a great deal of structurebeyond being positive semi-definite. It can always be split into two kernelspossessing certain shift invariance properties.

Kernel decompositions for Schur functions on the polydisk